GPU-Accelerated Particle Methods for Evaluation of Sparse Observations for PDE-Constrained Inverse Problems

TL;DR

This paper introduces GPU-accelerated particle methods for efficiently solving PDE-constrained inverse problems involving sparse observations, significantly improving computational speed for high-dimensional parameter estimation tasks.

Contribution

The paper presents two novel particle methods tailored for inverse PDE problems that leverage GPU acceleration to handle sparse data and high-dimensional unknowns efficiently.

Findings

Methods enable substantial speedup on modern hardware.

Applicable to Bayesian inference and numerical optimization.

Scalable to high-dimensional inverse problems.

Abstract

We consider the inverse problem of estimating parameters of a driven diffusion (e.g., the underlying fluid flow, diffusion coefficient, or source terms) from point measurements of a passive scalar (e.g., the concentration of a pollutant). We present two particle methods that leverage the structure of the inverse problem to enable efficient computation of the forward map, one for time evolution problems and one for a Dirichlet boundary-value problem. The methods scale in a natural fashion to modern computational architectures, enabling substantial speedup for applications involving sparse observations and high-dimensional unknowns. Numerical examples of applications to Bayesian inference and numerical optimization are provided.